mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
hF : Cauchy F
h0 : 𝓝 0 ⊓ F = ⊥
U : Set K
U_in : U ∈ 𝓝 0
M : Set K
M_in : M ∈ F
H : U ∩ M = ∅
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
hU : {x | v.restrict x < ↑γ₀} ⊆ U
⊢ ∀ x ∈ M, ↑γ₀ ≤ v.restrict x

Tactic:
intro x

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
hF : Cauchy F
h0 : 𝓝 0 ⊓ F = ⊥
U : Set K
U_in : U ∈ 𝓝 0
M : Set K
M_in : M ∈ F
H : U ∩ M = ∅
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
hU : {x | v.restrict x < ↑γ₀} ⊆ U
x : K
⊢ x ∈ M → ↑γ₀ ≤ v.restrict x
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: e : ℤ ≃+o ℤ
H : ↑e = AddEquiv.neg ℤ
⊢ False

Tactic:
replace H : e 1 = -1 := congr($H 1)
      have h1 : 0 < e 1 := by
        rw [← map_zero e, map_lt_map_iff]
        simp
      simp [H] at h1

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
⊢ Function.LeftInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀

Tactic:
intro x

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
x : hat K
⊢ (fun x ↦ x * x₀⁻¹) ((fun x ↦ x * x₀) x) = x
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
⊢ Function.RightInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀

Tactic:
intro x

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun x ↦ x * x₀⁻¹) fun x ↦ x * x₀
x : hat K
⊢ (fun x ↦ x * x₀) ((fun x ↦ x * x₀⁻¹) x) = x
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
h : 1 < b
ha : 1 ≤ b
⊢ b ∈ lowerBounds {y | y ∈ closure {b} ∧ 1 < y}

Tactic:
intro x

Goal state after:
case refine_2
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
h : 1 < b
ha : 1 ≤ b
x : G
⊢ x ∈ {y | y ∈ closure {b} ∧ 1 < y} → b ≤ x
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case neg
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
H : ∀ (x : G), ¬IsLeast {y | 0 < y} x
⊢ ∀ (a₁ a₂ : G), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂

Tactic:
intro x

Goal state after:
case neg
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
H : ∀ (x : G), ¬IsLeast {y | 0 < y} x
x : G
⊢ ∀ (a₂ : G), x < a₂ → ∃ a, x < a ∧ a < a₂
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case ih
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ ∀ (a b : K), extension (↑a * ↑b) = extension ↑a * extension ↑b

Tactic:
intro x

Goal state after:
case ih
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x✝ y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : K
⊢ ∀ (b : K), extension (↑x * ↑b) = extension ↑x * extension ↑b
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case ih
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ ∀ (a b : K), extension (↑a + ↑b) ≤ extension ↑a ∨ extension (↑a + ↑b) ≤ extension ↑b

Tactic:
intro x

Goal state after:
case ih
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x✝ y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : K
⊢ ∀ (b : K), extension (↑x + ↑b) ≤ extension ↑x ∨ extension (↑x + ↑b) ≤ extension ↑b
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_1
G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
hg' : g⁻¹ ≠ 0
h : ({x | x ≤ g} \ {0}).WellFoundedOn fun x1 x2 ↦ x1 > x2
⊢ MapsTo (fun x ↦ x⁻¹) {x | g⁻¹ ≤ x} ({x | x ≤ g} \ {0})

Tactic:
intro x

Goal state after:
case refine_1
G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
hg' : g⁻¹ ≠ 0
h : ({x | x ≤ g} \ {0}).WellFoundedOn fun x1 x2 ↦ x1 > x2
x : G₀
⊢ x ∈ {x | g⁻¹ ≤ x} → (fun x ↦ x⁻¹) x ∈ {x | x ≤ g} \ {0}
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_3
G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
hg' : g⁻¹ ≠ 0
h : {x | g⁻¹ ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
⊢ MapsTo (fun x ↦ x⁻¹) ({x | x ≤ g} \ {0}) {x | g⁻¹ ≤ x}

Tactic:
intro x

Goal state after:
case refine_3
G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
hg' : g⁻¹ ≠ 0
h : {x | g⁻¹ ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
x : G₀
⊢ x ∈ {x | x ≤ g} \ {0} → (fun x ↦ x⁻¹) x ∈ {x | g⁻¹ ≤ x}
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : CommMonoid α
inst✝³ : PartialOrder α
inst✝² : CommMonoid β
inst✝¹ : PartialOrder β
e : α ≃*o β
inst✝ : MulArchimedean α
⊢ ∀ (x : β) {y : β}, 1 < y → ∃ n, x ≤ y ^ n

Tactic:
intro x

Goal state after:
α : Type u_2
β : Type u_3
inst✝⁴ : CommMonoid α
inst✝³ : PartialOrder α
inst✝² : CommMonoid β
inst✝¹ : PartialOrder β
e : α ≃*o β
inst✝ : MulArchimedean α
x : β
⊢ ∀ {y : β}, 1 < y → ∃ n, x ≤ y ^ n
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ₁ φ₂ φ₃ : MvPowerSeries σ R
⊢ φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃)

Tactic:
ext1 n

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff n) (φ₁ * φ₂ * φ₃) = (coeff n) (φ₁ * (φ₂ * φ₃))
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case neg.hnc
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : ¬n ≤ d
x : ℕ × ℕ
hx : x ∈ antidiagonal d
⊢ ¬x.2 = n

Tactic:
exact ((le_of_add_le_right (mem_antidiagonal.mp hx).le).trans_lt <| not_le.mp h).ne

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
S : Type u_2
inst✝ : Semiring S
f : R →+* S
r : R
n✝ : ℕ
⊢ (coeff n✝) ((map f) (C r)) = (coeff n✝) (C (f r))

Tactic:
simp [coeff_C, apply_ite f]

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
hab : extensionValuation.restrict ↑⋯.choose = extensionValuation.restrict ↑⋯.choose
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
hx : v x = embedding a
hy : v y = embedding b
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
ha0 : ¬a = 0
⊢ embedding a = embedding b

Tactic:
· apply_fun embedding at ha0 using embedding_injective (f := (.ofClass hv.v))
      have h0 : extension ((algebraMap K (hat K)) x) ≠ 0 := by
        simp only [ne_eq, extension_eq_zero_iff, map_eq_zero]
        intro h
        simp [h, ← hx] at ha0
      have h0' : extension ((algebraMap K (hat K)) y) ≠ 0 := by
        have hb0 : b ≠ 0 := by
          apply_fun embedding at hab using embedding_injective (f := (.ofClass hv.v))
          simp only [← hx_def, Valuation.embedding_restrict, extensionValuation_apply_coe,
            ← hy_def] at hab
          simpa [← hx, hab, hy] using ha0
        apply_fun embedding at hb0 using embedding_injective (f := (.ofClass hv.v))
        simp only [ne_eq, extension_eq_zero_iff, map_eq_zero]
        intro h
        simp [h, ← hy] at hb0
      simp only [map_eq_zero, h0, reduceDIte, h0', WithZero.coe_inj, Subtype.mk.injEq,
        Units.mk0_inj, embedding_inj] at this
      simp only [Completion.algebraMap_def, Algebra.algebraMap_self, RingHom.id_apply,
        extension_extends, Valuation.restrict_inj] at this
      rwa [← hx, ← hy]

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case refine_2
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Function.Surjective ⇑valueGroup₀_hom_extensionValuation

Tactic:
intro x

Goal state after:
case refine_2
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : (ofClass extensionValuation).ValueGroup₀
⊢ ∃ a, valueGroup₀_hom_extensionValuation a = x
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : 1 ≤ n'
| constantCoeff φ ^ n' * (coeff 1) φ + constantCoeff φ ^ (n' - 1) * constantCoeff φ * ↑n' * (coeff 1) φ

Tactic:
2,

Goal state after:
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : 1 ≤ n'
| constantCoeff φ ^ (n' - 1) * constantCoeff φ * ↑n' * (coeff 1) φ
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : 1 ≤ n'
| constantCoeff φ ^ n' * (coeff 1) φ + constantCoeff φ ^ (n' - 1 + 1) * ↑n' * (coeff 1) φ

Tactic:
2,

Goal state after:
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
h' : 1 ≤ n'
| constantCoeff φ ^ (n' - 1 + 1) * ↑n' * (coeff 1) φ
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
⊢ Function.Surjective ⇑(valuation K v)

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : WithZero (Multiplicative ℤ)
⊢ ∃ a, (valuation K v) a = x
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := {i ∈ range n | k i = 0}
hs_def : s = {i ∈ range n | k i = 0}
hs : s ⊆ range n
hs' : ∀ i ∈ s, (coeff (k i)) f = constantCoeff f
hs'' : ∀ i ∈ s, k i = 0
this : m ≤ #s
⊢ constantCoeff f ^ #s = 0

Tactic:
by
    obtain ⟨m', hm'⟩ := Nat.exists_eq_add_of_le this
    rw [hm', pow_add, hf, zero_mul]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := {i ∈ range n | k i = 0}
hs_def : s = {i ∈ range n | k i = 0}
hs : s ⊆ range n
hs' : ∀ i ∈ s, (coeff (k i)) f = constantCoeff f
hs'' : ∀ i ∈ s, k i = 0
i : ℕ
hi : i < n
hi' : i < n → ¬k i = 0
⊢ ¬k i = 0

Tactic:
exact hi' hi

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B : Type u_4
inst✝³ : Semiring B
inst✝² : Algebra R B
inst✝¹ : Nonempty σ
inst✝ : Nontrivial R
inhabited_h : Inhabited σ
⊢ ∀ (x : R), ¬(algebraMap R (MvPowerSeries σ R)) x = X default

Tactic:
intro x

Goal state after:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B : Type u_4
inst✝³ : Semiring B
inst✝² : Algebra R B
inst✝¹ : Nonempty σ
inst✝ : Nontrivial R
inhabited_h : Inhabited σ
x : R
⊢ ¬(algebraMap R (MvPowerSeries σ R)) x = X default
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
n : ℕ
h : ∀ (d : σ →₀ ℕ), (weight w) d < n → (coeff d) f = 0
H : weightedOrder w f < ↑n
⊢ False

Tactic:
have : (f.weightedOrder w).toNat = f.weightedOrder w := by
    rw [natCast_toNat_eq_self]; exact ne_top_of_lt H

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
n : ℕ
h : ∀ (d : σ →₀ ℕ), (weight w) d < n → (coeff d) f = 0
H : weightedOrder w f < ↑n
this : ↑(weightedOrder w f).toNat = weightedOrder w f
⊢ False
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
⊢ Function.Injective toMvPowerSeries

Tactic:
intro x

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
x : MvPolynomial σ R
⊢ ∀ ⦃a₂ : MvPolynomial σ R⦄, ↑x = ↑a₂ → x = a₂
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : CommSemiring R
A : Type u_4
inst✝¹ : CommSemiring A
inst✝ : Algebra A R
e : σ →₀ ℕ
a : σ → A
⊢ ∀ x ∈ e.support,
    (a x • MvPowerSeries.X x) ^ e x = (MvPowerSeries.C ((algebraMap A R) (a x)) * MvPowerSeries.X x) ^ e x

Tactic:
intro x

Goal state after:
σ : Type u_1
R : Type u_2
inst✝² : CommSemiring R
A : Type u_4
inst✝¹ : CommSemiring A
inst✝ : Algebra A R
e : σ →₀ ℕ
a : σ → A
x : σ
⊢ x ∈ e.support → (a x • MvPowerSeries.X x) ^ e x = (MvPowerSeries.C ((algebraMap A R) (a x)) * MvPowerSeries.X x) ^ e x
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case h
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv : (valuation K v) x ≤ 1
γ : (WithZero (Multiplicative ℤ))ˣ
n d : R
hd✝ : d ∈ v.asIdeal.primeCompl
hd : v.intValuation d = 1
hnd : x * (algebraMap R K) ↑⟨d, hd✝⟩ = (algebraMap R K) n
hd' : v.intValuation n ≤ v.intValuation d
a : R
hval : v.intValuation (n - a * d) < ↑γ
⊢ (valuation K v) ((algebraMap R K) a - x) < ↑γ

Tactic:
suffices h : v.valuation K (algebraMap R K a - x) = v.intValuation (n - a * d) by rwa [h]

Goal state after:
case h
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv : (valuation K v) x ≤ 1
γ : (WithZero (Multiplicative ℤ))ˣ
n d : R
hd✝ : d ∈ v.asIdeal.primeCompl
hd : v.intValuation d = 1
hnd : x * (algebraMap R K) ↑⟨d, hd✝⟩ = (algebraMap R K) n
hd' : v.intValuation n ≤ v.intValuation d
a : R
hval : v.intValuation (n - a * d) < ↑γ
⊢ (valuation K v) ((algebraMap R K) a - x) = v.intValuation (n - a * d)
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : IsWeightedHomogeneous w f p
hg : IsWeightedHomogeneous w g q
d : σ →₀ ℕ
hd : ¬(weight w) d = p + q
⊢ ∀ x ∈ Finset.antidiagonal d, (coeff x.1) f * (coeff x.2) g = 0

Tactic:
intro x

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : IsWeightedHomogeneous w f p
hg : IsWeightedHomogeneous w g q
d : σ →₀ ℕ
hd : ¬(weight w) d = p + q
x : (σ →₀ ℕ) × (σ →₀ ℕ)
⊢ x ∈ Finset.antidiagonal d → (coeff x.1) f * (coeff x.2) g = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
hd : (weight w) d = p + q
⊢ ∀ x ∈ Finset.antidiagonal d,
    (coeff x.1) f * (coeff x.2) g =
      (coeff x.1) ((weightedHomogeneousComponent w p) f) * (coeff x.2) ((weightedHomogeneousComponent w q) g)

Tactic:
intro x

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
hd : (weight w) d = p + q
x : (σ →₀ ℕ) × (σ →₀ ℕ)
⊢ x ∈ Finset.antidiagonal d →
    (coeff x.1) f * (coeff x.2) g =
      (coeff x.1) ((weightedHomogeneousComponent w p) f) * (coeff x.2) ((weightedHomogeneousComponent w q) g)
Module: Mathlib.RingTheory.MvPowerSeries.PiTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
u : Filter ι
g : MvPowerSeries σ R
⊢ (∀ (i : σ →₀ ℕ), Tendsto (fun x ↦ f x i) u (𝓝 (g i))) ↔
    ∀ (d : σ →₀ ℕ), Tendsto (fun i ↦ (coeff d) (f i)) u (𝓝 ((coeff d) g))

Tactic:
exact forall_congr' (fun d => Iff.rfl)

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.PiTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
⊢ Continuous[inst✝¹, instTopologicalSpace R] ⇑C

Tactic:
classical
  simp only [continuous_iff_continuousAt]
  refine fun r ↦ (tendsto_iff_coeff_tendsto _ _ _).mpr fun d ↦ ?_
  simp only [coeff_C]
  split_ifs
  · exact tendsto_id
  · exact tendsto_const_nhds

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝⁴ : Field K
inst✝³ : CommSemiring S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
v : HeightOneSpectrum R
inst✝ : FaithfulSMul R K
⊢ Function.Injective ⇑(algebraMap R ↥(adicCompletionIntegers K v))

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝⁴ : Field K
inst✝³ : CommSemiring S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
v : HeightOneSpectrum R
inst✝ : FaithfulSMul R K
x : R
⊢ ∀ ⦃a₂ : R⦄, (algebraMap R ↥(adicCompletionIntegers K v)) x = (algebraMap R ↥(adicCompletionIntegers K v)) a₂ → x = a₂
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
⊢ φ.order = ↑n ↔ (coeff n) φ ≠ 0 ∧ ∀ i < n, (coeff i) φ = 0

Tactic:
rcases eq_or_ne φ 0 with (rfl | hφ)

Goal state after:
case inl
R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ order 0 = ↑n ↔ (coeff n) 0 ≠ 0 ∧ ∀ i < n, (coeff i) 0 = 0

case inr
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
hφ : φ ≠ 0
⊢ φ.order = ↑n ↔ (coeff n) φ ≠ 0 ∧ ∀ i < n, (coeff i) φ = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
⊢ IsNonarchimedean (v.intAdicAbvDef hb)

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
x : R
⊢ ∀ (b_1 : R), v.intAdicAbvDef hb (x + b_1) ≤ max (v.intAdicAbvDef hb x) (v.intAdicAbvDef hb b_1)
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
⊢ IsNonarchimedean (v.adicAbvDef hb)

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
x : K
⊢ ∀ (b_1 : K), v.adicAbvDef hb (x + b_1) ≤ max (v.adicAbvDef hb x) (v.adicAbvDef hb b_1)
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
φ ψ : R⟦X⟧
n : ℕ
h : ↑n < ψ.order
⊢ ∀ x ∈ antidiagonal n, (coeff x.1) φ * (coeff x.2) ψ = 0

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝ : Semiring R
φ ψ : R⟦X⟧
n : ℕ
h : ↑n < ψ.order
x : ℕ × ℕ
⊢ x ∈ antidiagonal n → (coeff x.1) φ * (coeff x.2) ψ = 0
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_2
inst✝ : Semiring R
φ : R⟦X⟧
⊢ φ.order = emultiplicity X φ

Tactic:
rcases eq_or_ne φ 0 with (rfl | hφ)

Goal state after:
case inl
R : Type u_2
inst✝ : Semiring R
⊢ order 0 = emultiplicity X 0

case inr
R : Type u_2
inst✝ : Semiring R
φ : R⟦X⟧
hφ : φ ≠ 0
⊢ φ.order = emultiplicity X φ
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
f g : R⟦X⟧
h : f ≠ 0 ∧ g ≠ 0
⊢ X ^ (f.order.toNat + g.order.toNat) * (f * g).divXPowOrder = X ^ (f * g).order.toNat * (f * g).divXPowOrder

Tactic:
by
        rw [order_mul, ENat.toNat_add (order_eq_top.not.mpr h.1) (order_eq_top.not.mpr h.2)]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: case mp.left
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
⊢ ↑J ⊆ ↑K

Tactic:
intro x

Goal state after:
case mp.left
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
x : R
⊢ x ∈ ↑J → x ∈ ↑K
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: case mp.right
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
h' : ¬e ≤ d
⊢ ⊤ ≤ K

Tactic:
intro x

Goal state after:
case mp.right
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
h' : ¬e ≤ d
x : R
⊢ x ∈ ⊤ → x ∈ K
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsLinearTopology R R
f : MvPowerSeries σ R
⊢ Tendsto (fun n ↦ f ^ n) atTop (𝓝 0) ↔ IsTopologicallyNilpotent (constantCoeff f)

Tactic:
by
  refine ⟨fun H ↦ ?_, isTopologicallyNilpotent_of_constantCoeff⟩
  replace H : Tendsto (fun n ↦ constantCoeff (f ^ n)) atTop (nhds 0) :=
    continuous_constantCoeff R |>.tendsto' 0 0 constantCoeff_zero |>.comp H
  simpa only [map_pow] using! H

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: c : ℝ
f g : ℕ → Bool
n : ℕ
⊢ Uncountable ℝ

Tactic:
rw [← aleph0_lt_mk_iff, mk_real]

Goal state after:
c : ℝ
f g : ℕ → Bool
n : ℕ
⊢ ℵ₀ < 𝔠